AROUND MATRIX-TREE THEOREM
YURII BURMAN AND BORIS SHAPIRO
Abstract. Generalizing the classical matrix-tree theorem we provide a formula counting subgraphs of a given graph with a fixed 2-core. We use this generalization to obtain an analog of the matrix-tree theorem for the root system
Dn (the classical theorem corresponds to the An -case). Several byproducts
of the developed technique, such as a new formula for a specialization of the
multivariate Tutte polynomial, are of independent interest.
1. Introduction
Let us first fix some definitions and notation to be used throughout the paper.
The main object of our study will be an undirected graph G without multiple edges.
It is understood as a subset G ⊂ {{i, j} | i, j ∈ {1, 2, . . . , n}}, where elements of
{1, 2, . . . , n} are vertices and elements of G itself are edges. Informally speaking,
this means that we mark (i.e. distinguish) vertices but not edges of G (except for
Section 6 where an edge labeling will be used). Usually we will assume that G
contains no loops, i.e. edges {i, i}. Directed graphs (appearing in Sections 2 and 5
for technical purposes) are subsets of {1, 2, . . . , n}2 . Since a graph is understood as
a set of edges, notation F ⊂ G means that F is a subgraph of G.
We will denote by n = v(G) the number of vertices of G, by #G = e(G) the
number of its edges, and by k(G) the number of connected components. For every
connected component Gi ⊂ G (i = 1, . . . , k(G)) it will be useful to consider its Euler
characteristics χ(Gi ) = v(Gi ) − e(Gi ). A connected graph containing no cycles will
be called a tree, a disconnected one, a forest. Note that the absence of cycles is
equivalent to the equality χ(Gi ) = 1 for all i; if cycles are present then χ(Gi ) ≤ 0.
We will usually supply edges of the graph G with weights. A weight wij = wji
of the edge {i, j} is an element of any algebra A. For a subgraph F ⊂ G denote
def Q
w(F ) = {i,j}∈F wij ; call it the weight of F . For any set U of subgraphs of G call
P
the expression Z(U ) = F ∈U w(F ) the statistical sum of U . (By definition, we
assume wij = 0 if G contains no edge {i, j}.)
To a graph G with weighted edges one associates its Laplacian matrix LG . It is
a symmetric (n × n)-matrix with the elements
(
−wij ,
i 6= j,
(LG )ij = P
k6=i wik , i = j.
The Laplacian matrix is degenerate; its kernel always contains the vector (1, 1, . . . , 1).
However, its principal minors are generally nonzero and enter the classical matrixtree theorem whose first version was proved by G. Kirchhoff in 1847:
1991 Mathematics Subject Classification. Primary 05C50, secondary 05B35.
Key words and phrases. Tutte polynomial, matrix-tree theorem, subgraph count.
Research supported in part by the RFBR grants # N.Sh.1972.2003.1 and # 05-01-01012a.
1
2
YU. BURMAN AND B. SHAPIRO
Theorem 1 ([7]). Let TG be the set of all (spanning) trees of G. Then Z(TG ) is
equal to any principal minor of LG .
This theorem has numerous generalizations (for a review, see e.g. [1] and the
references therein). For our purposes the most important will be the “all-minors”
theorem by S. Chaiken [2].
Call a subset J = {(i1 , j1 ), . . . , (im , jm )} ⊂ {1, 2, . . . , n}2 component-disjoint if
def Pm
ip 6= iq and jp 6= jq for every p 6= q; denote ΣJ = p=1 (ip + jp ). Fix a numeration
of the pairs (ip , jp ) ∈ J such that i1 < · · · < im , and denote by τJ the permutation
of {1, 2, . . . , m} defined by the condition jτJ (1) < jτJ (2) < · · · < jτJ (m) .
A forest F with the vertex set {1, 2, . . . , n} is called J-admissible if it has m components, and every component contains exactly one vertex from the set {i1 , . . . , im },
and exactly one, from {j1 , . . . , jm } (these two may coincide if the sets intersect).
Denote by γF,J a permutation of the set {1, 2, . . . , m} such that ip and jγF,J (p) lie
in the same component of F , for every p = 1, 2, . . . , m.
For an n × n-matrix M and a component-disjoint set J denote by M (J) the submatrix of M obtained by deletion of the rows i1 , . . . , im and the columns j1 , . . . , jm .
For any permutation σ denote by ε(σ) = ±1 its sign (parity).
Theorem 2 ([2]). For any component-disjoint subset J ⊂ {1, 2, . . . , n}2 one has
X
(−1)ΣJ det(LG )(J) =
ε(τJ ◦ γF,J )w(F )
F
where the sum is taken over the set of all J-admissible subforests F of G.
Theorem 1 is a particular case of Theorem 2 corresponding to the situation when
J contains one element only.
Most of this article is devoted to various generalizations of Theorem 1. In Section
2 we consider determinant-like expressions for statistical sums of subgraphs F ⊂ G
with cycles (namely, subgraphs with a given 2-core). In Section 3 we consider the
case of subgraphs with vanishing Euler characteristics. Spanning trees of a graph
G can be interpreted as irreducible linearly independent subsets of roots in the root
system An ; in Section 4 we prove an analog of Theorem 1 for the root system Dn .
Two remaining sections form a sort of appendix to the paper. They both deal
with the multivariate Tutte polynomial of the graph G defined as
n
X
TG (q, w) =
q n Z(Um )
m=1
(see [11, 15, 17] for details) where Um is the set of subgraphs of G having m
connected components and w is the collection of weights of the edges. The number
d(G) of totally cyclic orientations of the graph G (this number enters Theorem 3) is
known to be a special value of the Tutte polynomial; in Section 5 we provide some
more formulas for d(G). Section 6 contains a new formula for another specialization
of TG that we call the external activity polynomial. The formula is an (alternating
sign) summation over partitions of the set of vertices of G.
In the end of the paper we discuss several open problems related to the main
topic.
Acknowledgments. The first named author is sincerely grateful to the Mathematics Department of Stockholm University for the hospitality and financial support of his visit in September 2005 when the essential part of this project was
AROUND MATRIX-TREE THEOREM
3
carried out. We are thankful to Professors N. Alon and A. Sokal for their comments
on the Tutte polynomial and a number of relevant references.
2. Graphs with a given 2-core
Let G be an undirected graph (loops and multiple edges are allowed). The
maximal subgraph G0 ⊂ G such that every vertex of G0 is an endpoint of at least
two edges or is attached to a loop (that is, there are no “hanging” vertices) is called
the 2-core of G and denoted by core2 (G). A graph G is the union of core2 (G) and
a number of forests (possibly empty) attached to every vertex of core2 (G).
A graph G is called negative if it contains no loops, no multiple edges, G =
core2 (G), and χ(Gi ) < 0 where Gi , i = 1, . . . , k(G) are connected components
of G. A graph G is called non-positive if all the above is true but χ(Gi ) ≤ 0.
A non-positive graph G is the union of a negative graph G0 and several cycles,
each cycle forming a separate connected component. We will code this situation as
G = G0 ∪ 3k3 . . . nkn where ks stands for the total number of cycles of length s.
For any directed graph Q (with the vertex set {1, 2, . . . , n}) denote by [Q] the
corresponding undirected graph. Given a (n × n)-matrix M with entries ai,j ∈ A
define
Y
def
hM, Qi =
aij .
(i,j)∈Q
In particular, if [Q] ⊂ G where G is a graph without loops or multiple edges, with
weights wij (like in the previous section), then hLG , Qi = (−1)e(Q) w(Q).
A directed graph Q is called regular if the following two conditions are satisfied:
(1) Q contains no sources or sinks, i.e. for every vertex there is at least one
incoming and one outgoing edge.
(2) If Q contains a loop (an edge (i, i)) or a pair of antiparallel edges (edges
(i, j) and (j, i)) then they form a separate connected component of Q.
If Q is a regular directed graph then [Q] consists of a non-positive graph and several
loops and double edges (cycles of length 2), each loop and double edge forming a
separate connected component. We will denote this by [Q] = H ∪ 1k1 2k2 where H
is non-positive and k1 , k2 are the number of loops and double edges, respectively.
In what follows it will be convenient to allow graphs to have multiple (more
specifically, double) edges. If F is a graph with multiple edges we will abuse notation
writing F ⊂ G if the graph obtained from F by neglecting the multiplicities is a
subgraph of G. Computing the weights, we will, however, take multiplicities into
account:
Y
def
m
w(F ) =
wij ij
{i,j} is an edge of F
where mij ∈ Z≥0 is the multiplicity of the edge {i, j}.
Let H ⊂ G be a non-positive graph plus several double edges, each double edge
forming a separate component. In other words, H = H0 ∪ 2k2 3k3 . . . nkn where H0
is negative. Then denote
X
%G (H) =
hLG , Λi
Λ is regular
[Λ] = H ∪ 1n−v(H)
(so that the total number of vertices of Λ is n). By U (H) denote the set of all
subgraphs F ⊂ G such that core2 (F ) = H.
4
YU. BURMAN AND B. SHAPIRO
Theorem 3. Let H = H0 ∪ 2k2 3k3 . . . nkn be a non-positive graph without loops
together with several double edges. Then
(2.1)
n
n
X
X
l2
ln l3 +···+ln
e(H)
%G (H) = (−1)
d(H0 )
···
...
Z(U (H0 ∪2l2 . . . nln ))
2
k2
kn
l2 =k2
ln =kn
where d(H0 ) = TH0 (−1, 1) is the number of totally cyclic orientations of H0 (i.e.
orientations without sources and sinks).
Notation TH0 (−1, 1) means that we substitute q = −1 in the multivariate Tutte
polynomial and assume that wij = 1 for all i, j.
Corollary 1. One has
(2.2)
Z(U (H)) = (−1)e(H0 ) d(H0 )2−(k3 +···+kn ) ×
n
n
X
X
ln
l2 +l3 +···+ln l2
×
···
(−1)
...
%G (H0 ∪ 2l2 . . . nln ).
k2
kn
l2 =k2
ln =kn
Remark. Corollary 1 is our closest approximation to a “matrix-subgraph” theorem,
that is, the best available analog of Theorem 1 for subgraphs of arbitrary structure.
Indeed, the left-hand side of (2.2) is the statistical sum over the graphs with a
fixed 2-core (for trees the 2-core is empty), while the right-hand side is a polylinear
function of matrix elements of the Laplacian matrix (in the case of trees it was its
principal minor). Notice that, unlike Theorem 1, the right-hand side of (2.2) cannot
be computed in polynomial time. This is hardly surprising since the calculation of
the Tutte polynomial (and even its value at almost any point of the plane) is a sharp
P -hard problem (see [16, §9]). Therefore there is no hope to obtain a formula for
the statistical sum of connected subgraphs in G with any given number of edges in
the form of a determinant or, in general, to get a formula of polynomial complexity
in terms of the Laplacian matrix.
Proof of Theorem 3. Let Λ = Λ0 ∪Λ1 be a regular subgraph of G such that [Λ0 ] = H
and [Λ1 ] = 1n−v(H) . Now, hLG , Λi = hLG , Λ0 ihLG , Λ1 i. Since Λ0 contains no loops,
e(H)
then hLG , Λ0 i = (−1)P
w(H).
One has (LG )ii = k6=i wik , so that the term hLG , Λ1 i can be represented as the
sum of monomials w
Pi1 k1 . . . wis ks where {i1 , . . . , is } is the vertex set of Λ1 . In other
words, hLG , Λ1 i = Θ w(Θ) where Θ is the directed graph with [Θ] ⊂ G satisfying
the following property: if i ∈ {i1 , . . . , is } then Θ contains exactly one edge starting
from i, and if i ∈
/ {i1 , . . . , is } is a vertex of Θ then it is a sink (no edge starts from
it).
One can easily see that every connected component of Θ is either a tree such
that all its vertices except the root are in {i1 , . . . , is }, or a graph with exactly one
cycle with all its vertices in {i1 , . . . , is }. Thus, core2 ([Λ0 ∪ Θ]) = H0 ∪ 2l2 . . . nln ,
where l2 ≥ k2 , . . . , ln ≥ kn .
On the other hand, let F ⊂ G be a subgraph such that core2 (F ) = H0 ∪2l2 . . . nln .
To identify F with [Λ0 ∪ Θ] one has, first, to point out which “1-cycled”
connected
components of F belong to Λ0 and which to Θ — there are kl22 . . . klnn ways to do
this. Having this choice made one must orient the 2-core of F without sources and
sinks — the number of such orientations being d(H0 ∪ 2l2 . . . nln ) = 2l3 +···+ln d(H0 ).
See [17] for the proof of the equality d(H) = TH (−1, 1).
AROUND MATRIX-TREE THEOREM
Pl
1 dk
(1−x)l = s=0 ks sl (−1)s xs =
Proof of Corollary 1. One has k!
dxk
and therefore
(
l
X
0, if k < l,
l
s s
(−1)
=
k
s
1, if k = l.
s=0
5
l
k
(1−x)l−k ,
The corollary is now straightforward.
3. Graphs with vanishing Euler characteristics
Corollary 1 becomes particularly simple if H is a cycle. Namely, if H = s1
(a cycle of length s) then λG (H) is the statistical sum of the set of all connected
subgraphs F ⊂ G having exactly one cycle of length s. The “negative part” H0 of
the graph H is empty which implies d(H0 ) = 1.
Denote by Σn the symmetric group of order n acting on {1, 2, . . . , n}, and denote
by Dn the set of all partitions of n. For a permutation σ ∈ Σn having k1 cycles
def
of length 1, k2 cycles of length 2, etc., denote D(σ) = 1k1 . . . nkn ∈ Dn . Finally,
for any function f : D → A define the f -determinant of an (n × n)-matrix M with
entries ai,j ∈ A by the formula
X
detf (L) =
f (D(σ))a1,σ(1) . . . an,σ(n) .
σ∈Σn
Now one has
%G (2l2 . . . nln ) =
X
(−1)n+2l2 +···+nln (LG )1,σ(1) . . . (LG )n,σ(n)
D(σ)=1n−2l2 −···−nln 2l2 ...nln
= detχl2 ,...,ln LG ,
where
(
(−1)n+2l2 +···+nln , if k2 = l2 , k3 = l3 , . . . , kn = ln ,
χl2 ,...,ln (1k1 . . . nkn ) =
0,
otherwise.
Thus, Corollary 1 for a cycle takes the following form:
Statement 1. The statistical sum of the set of subgraphs F ⊂ G having one cycle
of length s ≥ 3 is equal to 21 detτs LG , where τs (1k1 . . . nkn ) = (−1)n+2k2 +···+nkn ks .
The statistical sum of the set of subgraphs F ⊂ G having one cycle of length 2 is
detτ2 LG .
This corollary implies the following formula which is the “matrix-tree theorem”
for connected subgraphs containing exactly one cycle of any length s ≥ 3, that is,
connected subgraphs H ⊂ G with χ(H) = 0:
Corollary 2. Let UG be the set of all connected subgraphs of H ⊂ G such that
χ(H) = 0. Then
1
Z(UG ) = detµ (LG )
2
where
(3.1)
µ(1k1 2k2 . . . nkn ) = (−1)n+k2 +2k3 +···+(n−1)kn (2k2 + k3 + · · · + kn ).
6
YU. BURMAN AND B. SHAPIRO
A finer result concerning graphs H ⊂ G such that χ(Hi ) = 0 for any connected
component Hi of H (i = 1, . . . , k(H)) can be obtained using Theorem 2.
For a graph G and a component-disjoint set J denote by G−J the graph obtained
from G by deletion of all the edges (ip jp ) where (ip , jp ) ∈ J. Then Theorem 2
implies
Statement 2. Let G be a graph with the vertex set {1, 2, . . . , n}, without loops and
multiple edges, with weights wij defined for all the edges. Let J = {(i1 , j1 ), . . . , (im , jm )}
be a component-disjoint subset of {1, 2, . . . , n}2 . Then
X
(3.2)
(−1)n ε(τJ )wi1 j1 . . . wim jm det(LG−J )(J) =
(−1)k(H) w(H)
H
where the sum is taken over the set of all subgraphs H ⊆ G such that every connected component Hi of H contains one cycle (that is, χ(Hi ) = 0), the edges
{i1 , j1 }, . . . , {im , jm } enter these cycles and vertices ip and jq alternate along the
cycle.
Proof. It follows from Theorem 2 that the product wi1 j1 . . . wim jm det(LG−J )(J) is
equal to the sum of ±wi1 j1 . . . wim jm w(F ) where F runs over the set of subforests of
G−J having m components and such that the p-th component contains the vertices
ip and jγF,J (p) ; here γF,J is the permutation of {1, 2, . . . , m} defined in Section 1.
In other words, wi1 j1 . . . wim jm det(LG−J )(J) is equal to the sum of ±w(H) where
H = F +J is the result of addition to F of the edges {i1 , j1 }, . . . , {im , jm }. Thus, H
is a graph with one cycle in every connected component; all edges {ip , jp } enter the
cycles, and vertices ip and jq alternate along the cycle. The connected components
of H are in one-to-one correspondence with the cycles of the permutation γF,J .
The sign of the term w(F ) is equal to (−1)n ε(τJ )ε(τJ ◦ γF,J ) = (−1)n ε(γG,J ). The
permutation γG,J contains k(H) cycles. The sign of any permutation of {1, 2, . . . , n}
with k cycles equals (−1)n+k , and therefore, the total sign is (−1)k(H) .
Denote now
Qm =
X
wi1 j1 . . . wim jm det(LG−J )(J)
#J=m
where the sum is taken over the set of all component-disjoint subsets J ⊂ {1, 2, . . . , n}2
of cardinality m. Statement 2 allows to express the generating function for the sequence Qm :
Theorem 4. One has
(3.3)
∞
X
m=1
k(H)
Qm t
m
n
= (−1)
X
H
w(H)
Y
((1 + t)`i (H) − 1)
i=1
where the sum in the right-hand side is taken over the set of all subgraphs H ⊂ G
such that core2 (Hi ) is a cycle of length li (H); here H1 , . . . , Hk(H) are connected
components of H.
P
Proof. By Statement 2 one has that Qm = H am (H)w(H) where the sum is taken
over the set of all subgraphs H ⊂ G having exactly one cycle in every connected
component. The coefficient am (H) is equal, to (−1)n+k(H) times the number of
component-disjoint sets J = {(i1 , j1 ), . . . , (im , jm )} such that
• {ip , jp } ∈ core2 (H) for all p = 1, . . . , m.
• For every cycle of H there is at least one edge (ip jp ) entering it.
AROUND MATRIX-TREE THEOREM
7
• If a cycle of H has more than one edge (ip , jp ) in it then the vertices ip and
jq alternate along the cycle.
This obviously implies that
n+k(H)
am (H) = (−1)
X
m1 + · · · + mk(H) = m
m1 , . . . , mk(H) ≥ 1
`1 (H)
`k(H) (H)
...
,
m1
mk(H)
and (3.3) follows.
Corollary 3.
(3.4)
∞
X
m=1
(−1)m Qm =
X
(−1)n+k(H) w(H).
H
4. Linearly independent subsets of the root systems An and Dn
The technique of Section 3 can be used to obtain results on linearly independent
subsets of finite root systems, cf. [10].
The set of positive roots R+ (An ) of the reflection group An consists of vectors
eij = bi − bj , 1 ≤ i < j ≤ n where b1 , . . . , bn is the standard basis in Cn . We will
assign to every root eij ∈ R+ (An ) its weight wij ∈ A where A is any algebra. By
definition wji = wij . For any subset S ⊂ R+ (An ) of positive roots consider a graph
Γ(S) with the vertices 1, . . . , n such that {i, j} is an edge of Γ(S) wherever eij ∈ S.
The edge {i, j} bears the weight wij . The graph Γ(S) is undirected and contains
no loops or multiple edges. If S 0 ⊂ S then Γ(S 0 ) is a subgraph of Γ(S). We will
write w(S) instead of w(Γ(S)) for short and denote by LS the Laplacian matrix of
the graph Γ(S).
For a given subset S ⊂ R+ (An ) one can consider the group G(S) generated by
the reflections in the roots eij ∈ S. The group
of the Weyl
PnG(S) is aPsubgroup
n
n
group of An , and therefore the space V = { i=1 xi bi |
i=1 xi = 0} ⊂ C is
G(S)-invariant. S is called irreducible if V is an irreducible representation of G(S).
The following is obvious:
Theorem 5. A set S ⊂ R+ (An ) is linearly independent if and only if Γ(S) contains
no cycles. S is irreducible if and only if Γ(S) is connected. A linearly independent
set S 0 ⊂ S is maximal (among linearly independent subsets of S) if and only if
Γ(S 0 ) is a forest composed of spanning trees of connected components of Γ(S). If
S is irreducible (that is, Γ(S) connected) then any maximal linearly independent
subset S 0 of S is also irreducible (that is, Γ(S 0 ) is a spanning tree of Γ(S)).
Using matroid terminology, one can reformulate Theorem 5 as follows. (See
[11, 17] for more detail about matroids.)
Corollary 4. A submatroid of the linear matroid of Cn generated by vectors eij ∈ S
is isomorphic to the graphical matroid of Γ(S).
One can associate a weight wij = wji ∈ A to every root eij ∈ R+ (An ). So,
one can consider weights of the root systems and statistical sums of sets of root
systems, like it was done for graphs in the previous sections. Now the matrix-tree
theorem (i.e. Theorem 1) and Theorem 5 imply:
8
YU. BURMAN AND B. SHAPIRO
Statement 3. Let S ⊂ R+ (An ) be irreducible and TS be the collection of all
maximal linearly independent subsets of S. Then Z(TS ) is equal to (any) principal
minor of the Laplacian matrix LS .
Consider now a similar question for the reflection group Dn . Its set R+ (Dn ) of
−
positive roots consists of the vectors e+
ij = bi −bj (the “+”-vectors) and eij = bi +bj
(the “–”-vectors) for all 1 ≤ i < j ≤ n. We associate to every “+”-vector e+
ij the
weight uij ∈ A, and to every “–”-vector e−
the
weight
v
∈
A.
Notions
of
linearly
ij
ij
independent, maximal and irreducible subsets S ⊂ R+ (Dn ) are defined exactly as
in the An -case.
For every set S ⊂ R+ (Dn ) consider the graph Γ(S) with the vertices 1, . . . , n
where the vertices i and j are joined by the edge marked “+” if e+
ij ∈ S, and by the
edge marked “–” if e−
∈
S.
Thus,
the
graph
Γ(S)
is
undirected,
contains no loops,
ij
and has at most two edges joining every pair of vertices; all its edges are marked
by “+” or “–”, and if two edges join the same pair of vertices then their marks are
different.
A cycle in Γ(S) is called odd if it contains an odd number of edges marked “–”.
Theorem 6. A set S ⊂ R+ (Dn ) is irreducible if and only if Γ(S) is connected. S
is linearly independent if and only if every connected component of Γ(S) is either a
tree or a graph with exactly one cycle, and this cycle is odd. If S is irreducible then
a linearly independent set S 0 ⊂ S is maximal if and only if the following holds: if
Γ(S) contains no odd cycles then S 0 = S, otherwise every connected component of
Γ(S 0 ) is a graph containing exactly one cycle, and this cycle is odd.
This is a D-analog of Theorem 5 and it is obvious as well. Our goal in this
section is to obtain a D-analog of Statement 3.
Let J = {(i1 , j1 ), . . . , (im , jm )} ⊂ {1, 2, . . . , n}2 be a component-disjoint subset.
def
−
Denote S − J − = S \ {e−
i1 j1 , . . . , eim jm }.
Theorem 7. One has
n
X
m=1
k(H)
t
m
X
n
vi1 j1 . . . vim jm det(LS−J − )(J) = (−1)
X
H
#J=m
w(H)
Y
−
((1+t)`i
(H)
−1).
i=1
Here the internal sum in the left-hand side is taken over the set of all componentdisjoint sets J ⊂ {1, 2, . . . , n}2 of cardinality m. The sum in the right-hand side
is taken over the set of all subsets H ⊂ S such that every connected component of
def
the graph Γ(H) contains exactly one cycle. Above we denote by k(H) = k(Γ(H))
−
the number of these components, and by `i (H) (i = 1, . . . , k(H)) the number of
“–”-edges entering the cycle in the i-th component.
The proof is completely analogous to that of Theorem 4. A required analog of
Statement 3 is now:
Corollary 5.
n
X
X
X
(4.1)
(−2)m
vi1 j1 . . . vim jm det(LS−J − )(J) = (−1)n
(−2)k(F ) w(F ).
m=1
#J=m
F
Here the internal sum in the left-hand side is taken over the set of all componentdisjoint sets J ⊂ {1, 2, . . . , n}2 of cardinality m. The sum in the right-hand side is
AROUND MATRIX-TREE THEOREM
9
taken over the set of all maximal linearly independent subsets F ⊂ S. As usual,
k(F ) is the number of connected components of the graph Γ(F ).
Proof. This follows directly from Theorems 7 and 6 and the equality
(
k
Y
(−2)k , if all the `i are odd,
((−1)`i − 1) =
0,
if at least one `i is even.
i=1
5. Totally cyclic orientations
Let G be an undirected graph with the vertex set {1, 2, . . . , n}, without loops
(multiple edges are allowed). In this section we give a combinatorial description of
the number d(G) of directed graphs Q such that [Q] = G and Q has no sources or
sinks. (Recall that the number d(G) enters equation (2.1).)
The classical formula for d(G)
X
d(G) = TG (−1, 1) =
(−1)k(F )
F ⊂G
was earlier quoted in Theorem 3. (See [11, 17] for a comprehensive review of
numerous other specializations of the Tutte polynomial.)
For a set of vertices P ⊂ {1, 2, . . . , n} of G denote by hP i the subgraph of G
spanned by P (i.e. having P as its vertex set and containing all the edges of G
def
with both endpoints in P ). Denote k(P ) = k(hP i) for short and denote by µ(P ) =
e(h{1, 2, . . . , n} \ P i), that is, the number of edges in G having both endpoints
outside P .
Recall that a graph F is called bipartite if one can split its vertices into two
groups such that every edge joins two vertices from different groups. Equivalently,
this means that every closed path in F contains an even number of edges.
Theorem 8. Let G have no isolated vertices. Then the number d(G) of totally
cyclic orientations of G (i.e. such that for every vertex there is at least one incoming
and one outgoing edge) is given by the expression
d(G) =
n
X
m=0
(−1)m
X
2µ(P )+k(P ) .
P :#P =m,hP i is bipartite
(By assumption, k(∅) = 0 and µ(∅) is equal to the total number of edges in G.)
Proof. Fix a set P of vertices, and let N (P ) be the number of orientations of G
such that every vertex from P is either a source or a sink. Since an edge cannot
join two sources or two sinks, one has N (P ) = 0 if hP i is not bipartite.
Suppose now that hP i is bipartite. Consider the graph F obtained by adding
to hP i all the edges having one vertex in P and the other outside P . Apparently,
k(F ) = k(P ). Since G has no isolated vertices, every connected component of F
has 2 orientations such that every its vertex is either a source or a sink. Thus,
the total number of ways to orient the edges of F is 2k(P ) . The number of edges
of G not belonging to F is µ(P ). These edges can be oriented arbitrarily, and,
therefore, N (P ) = 2µ(P )+k(P ) . The statement follows now from the inclusionexclusion formula.
10
YU. BURMAN AND B. SHAPIRO
Corollary 6. The number of totally cyclic orientations of G is given by
n
X
X
(5.1)
d(G) =
(−1)m
2µ(P ) chrhP i (2)
m=0
P :#P =m
where chrF (λ) is the chromatic polynomial of the graph F , that is, the number of
ways to color its vertices in λ colors so that any two adjacent vertices have different
colors. (One assumes chrh∅i = 1.)
Proof. One has chrF (2) = 2k(F ) if the graph F is bipartite, and chrF (2) = 0
otherwise.
Corollary 7. The number of totally cyclic orientations of G is given by
X
(5.2)
d(G) =
2µ(F )+k(F ) (−1)χ(F )
F ⊆G
where the sum is taken over the set of all subgraphs F ⊆ G, and µ(F ) is the total
number of edges in G having no common vertices with the edges from F .
Proof. A classical result (see e.g. [17] for proof) relates the multivariate Tutte polynomial to the chromatic polynomial:
X
chrH (λ) = TH (λ, −1) =
λk(F ) (−1)e(F ) .
F ⊆H
v(F )=v(H)
(−1 in the argument of TH means that one takes wij = −1 for every edge {i, j} of
H). Now by Corollary 6,
n
X
X
X
d(G) =
(−1)m
2µ(F ) 2k(F ) (−1)e(F ) =
2µ(F )+k(F ) (−1)χ(F ) .
m=0
F ⊆G
F ⊆G
v(F )=m
where χ(F ) = v(F ) − e(F ) is the Euler characteristics of F .
6. Multivariable external activity polynomial
In the previous section we made use of the fact that the chromatic polynomial
of a graph is a specialization of its multivariate Tutte polynomial T (q, w). Below
we consider another specialization of the T which we call the external activity
polynomial.
As in Section 2 let G be a graph without loops or multiple edges and with the
weights wij ∈ A assigned to its edges. Suppose that G is connected and fix an
arbitrary numeration of its edges. Let T be a spanning tree of G and e be an edge
not entering T . The graph T ∪ e has exactly one cycle, and this cycle contains the
edge e. An edge e is called externally active for T if e is the smallest edge (with
respect to the above numeration) in the cycle. The polynomial
X
Y
CG (w) =
w(T )
(wij + 1).
T is a spanning tree of G
{i,j} is externally active for T
will be called the external activity polynomial of G. Its specialization (re)appeared
recently in the form of the Hilbert polynomial for a certain commutative algebra
related to G, see [9].
AROUND MATRIX-TREE THEOREM
11
Obviously, the following statement holds:
Statement 4. If G is connected then CG (w) = Z(U0 ), where U0 is the set of all
connected spanning subgraphs of G.
Corollary 8 ([11]). If G is connected then CG (w) = limq→0 TG (q, w)/q where TG
is the multivariate Tutte polynomial.
We present another expression for the polynomial CG :
Theorem 9.
(6.1)
CG (w) =
n
X
X
(−1)k−1 (k − 1)!
Y
(wij + 1)
{1,2,...,n}=P1 t···tPk {i,j}:∃s i,j∈Ps
k=1
where the internal sum taken is over all partitions of the set of vertices into k ≥ 1
pairwise disjoint subsets, and the product is taken over the set of all edges {i, j} of
the graph G such that both endpoints (i and j) belong to some Ps , s = 1, . . . , k.
To prove Theorem 9 we need the following technical lemma.
Lemma 1. For any n ≥ 2 one has
n
X
(6.2)
X
(−1)k−1
(p1 − 1)! . . . (pk − 1)! = 0,
{1,2,...,n}=P1 t···tPk
k=1
where pi = #Pi is the cardinality of Pi .
Proof. Denote by ak,n the coefficient at (−1)k−1 (k − 1)! in (6.2) and use induction
on n to prove the lemma. For n = 2 one has a1,2 = 1 (the only possible set
partition is {1, 2} = {1, 2}) and a2,2 = 1 (the only possible set partition is {1, 2} =
{1} t {2}), so that (6.2) holds. Let now it hold for some n. Partitions of the set
{1, 2, . . . , n + 1} fall into two types: either {1, 2, . . . , n + 1} = P1 t · · · t Pk t {n + 1},
or {1, 2, . . . , n + 1} = P1 t · · · t (Ps ∪ {n + 1}) t · · · t Pk , where in both cases
P1 t · · · t Pk is a partition of {1, 2, . . . , n}. The sum taken over the set partitions
of the first type is
n
X
X
(−1)k
k=1
(p1 − 1)! . . . (pk − 1)! 0! = 0
{1,2,...,n+1}=P1 t···tPk t{n+1}
by the induction hypothesis. The sum taken over the set partitions of the second
type equals
n
X
k−1
(−1)
k
X
X
(p1 − 1)! . . . ps ! . . . (pk − 1)!
s=1 {1,2,...,n+1}=P1 t···t(Ps ∪{n+1})t···tPk
k=1
=
k
X
(−1)k−1
s=1
=n
k
X
s=1
k
X
s=1
(−1)k−1
X
ps
(p1 − 1)! . . . (ps − 1)! . . . (pk − 1)!
{1,2,...,n}=P1 t···tPs t···tPk
X
(p1 − 1)! . . . (pk − 1)!
{1,2,...,n}=P1 t···tPk
= 0.
12
YU. BURMAN AND B. SHAPIRO
Proof of Theorem 9. Notice first that for any graph F the statistical sum of all
subgraphs H ⊂ F equals
Y
Z({H | H ⊂ F }) =
(1 + wij ).
{i,j} is an edge of F
Consider the poset Pn of all set partitions of {1, 2, . . . , n} ordered by refinement.
In particular, min = {1} t · · · t {n} is the smallest element of Pn , and Max =
{1, 2, . . . , n} is its largest element.
Recall (from Section 5) that for a set P ⊂ {1, 2, . . . , n} one denotes by hP i the
subgraph of G with the vertex set P ; edges of hP i are all the edges of G with both
endpoints in P . Statement 4 implies now that
X
Z({H | H ⊂ F }) =
ChP1 i (w) . . . ChPk i (w).
(P1 t···tPk )∈Pn
By Lemma 1 the value µ(u, Max) = (−1)k−1 (k − 1)! where µ(u, v) is the Möbius
function for the poset Pn . Therefore theorem follows from the Möbius inversion
formula, see e.g. [12].
Questions and final remarks
Multivariate Tutte polynomials have been studied intensively since 1970s when
their relations with partition functions of several important models in mathematical
physics (e.g. Ising model, Potts model) were discovered. (For more information
consult [5, 3], references therein and also the review paper [11].) Of particular
interest are the complex zeros of these polynomials since they are responsible for
the phase transition in ferromagnetic and antiferromagnetic media. It might be
very interesting to study the zeros of the external activity polynomial GG . Many
natural questions about them (including the half-plane property, see [11]) are still
open. Notice however that by Corollary 8 the polynomial CG is related to the
specialization of the Tutte polynomial at q = 0 while in Potts model q is interpreted
as a number of states (of the spin) — we are leaving it to professional physicists to
give a sensible interpretation to the polynomial CG .
Another possible direction of study is suggested by the nature of formulas (2.1),
(2.2), (3.2), (3.4), (4.1), (5.1), (5.2) and (6.1): they all contain a sign alternating
summation. This makes us to suspect that these formulas present the Euler characteristics of suitable complexes and pose a problem to find these complexes, i.e. to
perform a “categorification”. A categorification of the chromatic polynomial was
recently carried out in [6, 13, 14].
Section 3 of this paper contains two different descriptions of graphs G such that
every connected component of G has vanishing Euler characteristics. One is given
by the Statement 1 and its corollary (both are special cases of Theorem 3), and the
other is contained in Theorem 4 and its corollary, based on the all-minors version of
the matrix-tree theorem. The relation between these results resembles the relation
between a determinant and its minors decomposition. It seems very interesting to
find similar results for graphs with an arbitrary 2-core.
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Independent University of Moscow, Bolshoi Vlas’evskii per. 11, 121002, Moscow,
Russia
E-mail address: [email protected]
Mathematics Department, Stockholm University, S-106 91 Stockholm, Sweden
E-mail address: [email protected]